Astron. Astrophys. 322, 730-746 (1997)

1. Introduction

About ten years ago we published a paper on the Malmquist bias and the value of the Hubble constant, as revealed by the Tully-Fisher (TF) relation (Bottinelli et al. 1986, or BGPT86). In that paper the method of normalized distances was for the first time applied to a sample of galaxies in an attempt to show the very existence of and to correct for the Malmquist bias when the value of is derived from field galaxies using the direct TF relation. The method accomplishes simultaneously these two tasks via the vs. diagram, where H is the Hubble ratio ( corrected radial velocity, TF distance from the direct TF relation) and is the normalized kinematical distance (we shall return to the details in Sect. 3).

BGPT86 illustrated in a clear manner how the Malmquist bias influences the inferred value of . The value of was derived from the "unbiased plateau" of the vs. diagram, using the calibrators then available. With two choices of calibrator distances we calculated km s^-1  Mpc^-1 and km s^-1  Mpc^-1, where the former value refers to the "short" (or de Vaucouleurs) distances and the latter one to the "long" (or Sandage) distances, in use at that time. It is the aim of the present paper to "update" in a very basic manner the results of BGPT86, after ten years of efforts by us and others to increase the data base, enhance the knowledge of the TF relation, and deepen the understanding of the Malmquist bias.

After BGPT86 it became clear that in order to put the investigation of and the local kinematics on a safer basis, when using the basic (or ) photometric parameters in the TF relation, one needs a significantly larger sample than available in 1986. There are several reasons for this. Firstly, the method of normalized distances requires complete samples, and to extend the unbiased plateau requires fainter photometric limit. Secondly, increasing the population of the unbiased plateau adds weight on from increasingly large distances, hence reducing the influence of the very local kinematical environment. Thirdly, the larger number of galaxies allows one to investigate in detail the properties of the sample, its completeness, and needed photometric corrections. Fourthly, similarly, the large sample makes it possible to study the TF-relation itself. Fifthly, though our main concern is the direct relation, any attempt to use the inverse TF relation (producing in a certain sense unbiased distance moduli) requires a good, homogeneous and well-understood photometric sample, along with the complete measurements of .

The roots of this work go back to 1983 when a plan was started to build an extragalactic database (Paturel et al. 1990) where the basic available measurements would be collected. This database (LEDA = Lyon-Meudon extragalactic database; telnet lmc.univ-lyon1.fr, login: leda) contains presently 140000 galaxies which have been used to homogenize the data relevant for TF studies. This part of the work has been reported by Bottinelli et al. (1990), Paturel et al. (1991a), and Paturel et al. (1994a), concerning 21-cm line measurements (), apparent diameters, and total apparent magnitudes, respectively.

In the project "Kinematics of the local universe" (KLUN), we have constructed a large sample of TF suitable galaxies, complete down to a small apparent diameter (Paturel et al. 1990). We selected diameter instead of magnitude as the defining parameter, because in this way it was possible to create with tolerable effort a deep sample, and because diameters also obey a TF relation. In fact, as a very useful byproduct of this project, it was also possible to make the magnitude limit fainter within the basic diameter-limited sample. Completing the sample required several hundreds of H I line profile measurements with the Nançay radiotelescope and hundreds of optical redshift measurements at Observatoire de Haute Provence and at ESO, La Silla, as a Key Project.

Preparations for the KLUN analysis have included a study of the inclination corrections by Bottinelli et al. (1995), where it was shown that the isophotal diameter changes very little when the viewing angle is changed (consistent with a high optical thickness of spiral galaxy disks at the 25 mag isophote), while B -magnitudes need a large correction. Another important preparative study concerned the type dependence of the TF relation. It was revealed especially using the inverse TF relations in Theureau et al. (1997), but it exists as well in the direct relations, both in diameters and magnitudes.

The Malmquist bias has received increasing attention during the last ten years, with advancement in our understanding of its different forms and situations in which it enters the results. Teerikorpi (1995) recommended the use of the terms "Malmquist bias of the 1st kind" and "Malmquist bias of the 2nd kind", in order to make a clear difference between two approaches in the studies using distance measurements. The bias of the 2nd kind is the one that arises in the analysis of the vs. diagram and which we try to overcome using the method of normalized distances. It is also the bias that Sandage (1994a, b) has discussed with his approach via the Spaenhauer diagram. The bias of the 1st kind is closely akin to the classical Malmquist bias, and has appeared in discussions on the "general" Malmquist correction.

Our approach to relies on a sufficiently good velocity field model that is used to calculate (relative) kinematical distances. In 1986 we used the Peebles linear model, centered around the Virgo cluster, for this purpose. Even though the years have brought about new information on possible large scale bulk flows (starting from the Hydra Centaurus flow as suggested by Tammann & Sandage 1985) and the possible Great Attractor infall flow (Lynden-Bell et al. 1988), we shall use the same basic kinematical model also in this study, which allows a comparison with the results of BGPT86.

Presently, the number of local calibrators with distances from primary indicators is significantly larger than in 1986. Our complete list contains now 30 local galaxies, half of them having primary Cepheid measurements, the other half with distances determined by group membership. It should be noted that 11 of the new primary (Cepheid) distances come from the HST programmes.

We outline below the structure of this paper:

In Sect. 2, the basic KLUN sample is described. The criteria for excluding parts of the sample due to problems of galactic extinction, kinematical closeness of the Virgo cluster, and inaccurate photometric parameters, are explained.

In Sect. 3, the method of normalized kinematical distances is explained, with some improvements in comparison with BGPT86. The comparable method of the Spaenhauer diagram (Sandage, 1994a) is discussed.

In Sect. 4, possible calibrators are discussed. We have decided to give two choices of calibrator sample: a smaller primary sample of galaxies with Cepheid distances, and a sample enlarged with group members.

In Sect. 5, we use both the diameter-limited sample, and the magnitude-limited subsample, in the iterative normalized distance method. We derive the slope of the TF relations, zero-point differences for the Hubble types, and the unbiased plateau in the vs. diagram. Systematic trends inside the putative plateaus, defined by different -limits, are studied. All this can be done without a knowledge of (or of calibrators).

In Sect. 6, we combine the data on calibrators with the results on the unbiased plateaus from Sect. 5, in order to derive the value of the Hubble constant. The calculations are made with different calibrator samples, and with different plateaus from Sect. 5.

Sect. 7 contains a discussion of the results on . We investigate the effect of the adopted velocity model, absorption correction, and absolute calibration. We conclude by comparing our results with recent SNIa studies.

© European Southern Observatory (ESO) 1997

Online publication: June 5, 1998

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